Extensions of bundles of null directions

The geometry of , the bundle of null directions over an Einstein spacetime, is studied. The full set of invariants of the natural G-structure on is constructed using the Cartan method of equivalence. This leads to an extension of which is an elliptic fibration over the spacetime. Examples are given which show that such an extension, although natural, is not unique. A reinterpretation of the Petrov classification in terms of the fibres of an extension of is presented.